Timelike infinity and asymptotic symmetry

Abstract

By extending Ashtekar and Romano’s definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike infinity for an isolated system with a source is proposed. The treatment provides unit spacelike three-hyperboloid timelike infinity and avoids the introduction of the troublesome differentiability conditions which were necessary in the previous works on asymptotically flat space–times at timelike infinity. Asymptotic flatness is characterized by the falloff rate of the energy-momentum tensor at timelike infinity, which makes it easier to understand physically what space–times are investigated. The notion of the order of the asymptotic flatness is naturally introduced from the rate. The definition gives a systematized picture of hierarchy in the asymptotic structure, which was not clear in the previous works. It is found that if the energy-momentum tensor falls off at a rate faster than ∼t−2, the space–time is asymptotically flat and asymptotically stationary in the sense that the Lie derivative of the metric with respect to ∂t falls off at the rate ∼t−2. It also admits an asymptotic symmetry group similar to the Poincaré group. If the energy-momentum tensor falls off at a rate faster than ∼t−3, the four-momentum of a space–time may be defined. On the other hand, angular momentum is defined only for space–times in which the energy-momentum tensor falls off at a rate faster than ∼t−4.